The idle period of the finite G/M/1 queue with an interpretation in risk theory

Löpker, Andreas; Perry, David

doi:10.1007/s11134-010-9168-z

“…By interpreting the interarrival times of the claims as service times of the corresponding queue and the claim sizes as interarrival times of the queue, the standard Cramér-Lundberg model is translated into a G/M/1 queue. The time to ruin in the Cramér-Lundberg model is now related to a busy period of the corresponding queue, the deficit at ruin to an idle period, and the surplus just before ruin to the sojourn time of the last customer in a busy period (see [16] and [19]).…”

Section: Introductionmentioning

confidence: 99%

Queues and Risk Models with Simultaneous Arrivals

Badila¹,

Boxma²,

Resing

³

et al. 2014

Advances in Applied Probability

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We focus on a particular connection between queueing and risk models in a multidimensional setting. We first consider the joint workload process in a queueing model with parallel queues and simultaneous arrivals at the queues. For the case that the service times are ordered (from largest in the first queue to smallest in the last queue), we obtain the Laplace-Stieltjes transform of the joint stationary workload distribution. Using a multivariate duality argument between queueing and risk models, this also gives the Laplace transform of the survival probability of all books in a multivariate risk model with simultaneous claim arrivals and the same ordering between claim sizes. Other features of the paper include a stochastic decomposition result for the workload vector, and an outline of how the two-dimensional risk model with a general two-dimensional claim size distribution (hence, without ordering of claim sizes) is related to a known Riemann boundary-value problem.

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“…In the GI/G/1 case, queueing model 1 corresponds to the standard finite dam with constant release rule; various versions have been treated in [2], [3], [6], [10], [11], [12], [21], [24], and [27]. From the point of view of prospective customers, the two models also represent GI/G/1 with deterministic customer patience.…”

Section: Queueingmentioning

confidence: 99%

A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

Perry

¹

,

Stadje

²

,

Zacks

³

2013

Journal of Applied Probability

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Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain 'dual' M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

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“…In the GI/G/1 case, queueing model 1 corresponds to the standard finite dam with constant release rule; various versions have been treated in [2], [3], [6], [10], [11], [12], [21], [24], and [27]. From the point of view of prospective customers, the two models also represent GI/G/1 with deterministic customer patience.…”

Section: Queueingmentioning

confidence: 99%

A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

Perry¹,

Stadje²,

Zacks³

2013

J. Appl. Probab.

2

0

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Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain 'dual' M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

show abstract

The idle period of the finite G/M/1 queue with an interpretation in risk theory

Cited by 11 publications

References 31 publications

Queues and Risk Models with Simultaneous Arrivals

Queues and Risk Models with Simultaneous Arrivals

A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

A Duality Approach to Queues with Service Restrictions and Storage Systems with State-Dependent Rates

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